Integration

Using the definition of integration where g is a step function supported on [a,b], (f continuous on [a,b] )

show that for .

I would really appreciate any help with this. It's easy to show that if lambda is 0 you get the result but I am having trouble proving the rest. I just get tangled up in algebra, though I think I am trying to prove sup A = sup B, where A = where g is a step function supported on [a,b], and B = where g is a step function supported on [a,b], Thank you very much.

provide :
for any step function g which supported on [a,b], , is a step function supported on [a,b], and .
taking the superem over all step function g, we get that the LHS is no less than the RHS.
and similarly the RHS is no less than the LHS. QED